Artin gluing a sheaf 4: a single sheaf in two ways

The goal of this post is to give an alternative perspective on making a sheaf over $X = \Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$, alternative to that of a previous post ("Artin gluing a sheaf 3: the Ran space," 2018-02-05). We will have one unique sheaf on all of $X$, valued either in simplicial complexes or simplicial sets.

Remark: Here we straddle the geometric category $SC$ of simplicial complexes and the algebraic category $\sSet$ of simplicial sets. There is a functor $[\ \cdot\ ]:SC\to \sSet$ for which every $n$-simplex in $S$ gets $(n+1)!$ elements in $[S]$, representing all the ways of ordering the vertices of $S$ (which we would like to view as unordered, to begin with).

Recall from previous posts:

• maps $f:X\to SC$ and $g = [f]:X\to \sSet$,
• the $SC_k$-stratification of $\Ran^k(M)\times \R_{\geqslant 0}$,
• the point-counting stratification of $\Ran^{\leqslant n}(M)$,
• the combined (via the product order) $SC_{\leqslant n}$-stratification of $\Ran^{\leqslant n}(M)\times \R_{\geqslant 0}$,
• an induced (by the $SC_k$-stratification) cover by nested open sets $B_{k,1},\dots,B_{k,N_k}$ of $\Ran^k(M)\times \R_{\geqslant 0}$,
• a corresponding induced total order $S_{k,1},\dots,S_{k,N_k}$ on $f(\Ran^k(M)\times \R_{\geqslant0})$.

The product order also induces a cover by nested opens of all of $X$ and a total order on $f(X)$ and $g(X)$. We call a path $\gamma:I\to X$ a descending path if $t_1<t_2\in I$ implies $h(\gamma(t_1))\geqslant h(\gamma(t_2))$ in any stratified space $h:X\to A$. Below, $h$ is either $f$ or $g$.

Lemma: A descending path $\gamma:I\to X$ induces a unique morphism $h(\gamma(0))\to h(\gamma(1))$.

Proof: Write $\gamma(0) = \{P_1,\dots,P_n\}$ and $\gamma(1) = \{Q_1,\dots,Q_m\}$, with $m\leqslant n$. Since the path is descending, points can only collide, not split. Hence $\gamma$ induces $n$ paths $\gamma_i:I\to M$ for $i=1,\dots,n$, with $\gamma_i$ the path based at $P_i$. This induces a map $h(\gamma(0))_0\to h(\gamma(1))_0$ on 0-cells (vertices or 0-objects), which completely defines a map $h(\gamma(0))\to h(\gamma(1))$ in the desired category. $\square$

Our sheaves will be defined using colimits. Fortunately, both $SC$ and $\sSet$ have (small) colimits. Finally, we also need an auxiliary function $\sigma:\Op(X)\to SC$ that finds the correct simplicial complex. Define it by \[ \sigma(U)  = \begin{cases}
S_{k,\ell} & \text{ if } U\neq\emptyset, \text{ for } k = \max\{1\leqslant k'\leqslant n\ :\ U\cap \Ran^k(M)\times \R_{\geqslant 0}\neq \emptyset\}, \\ & \hspace{2.23cm} \ell = \max\{1\leqslant \ell'\leqslant N_k\ :\ U \cap B_{k,\ell'}\neq\emptyset\},\\
* & \text{ if }U= \emptyset.
\end{cases} \]

Proposition 1: Let $\mathcal F$ be the function $\Op(X)^{op}\to SC$ on objects given by \[ \mathcal F(U) = \colim\left(\sigma(U)\rightrightarrows S\ :\ \text{every }\sigma(U)\to S \text{ is induced by a descending }\gamma:I\to U\right). \] This is a functor and satisfies the sheaf gluing conditions.

Proof: We have a well-defined function, so we have to describe the restriction maps and show gluing works. Since $V\subseteq U\subseteq X$, every $S$ in the directed system defining $\mathcal F(V)$ is contained in the directed system defining $\mathcal F(U)$. As there are maps $\sigma(V)\to \mathcal F(V)$ and $S\to \mathcal F(V)$, for every $S$ in the directed system of $V$, precomposing with any descending path we get maps $\sigma (U)\to \mathcal F(V)$ and $S\to \mathcal F(V)$, for every $S$ in the directed system of $U$. Then universality of the colimit gives us a unique map $\mathcal F(U)\to \mathcal F(V)$. Note that if there are no paths (decending or otherwise) from $U$ to $V$, then the colimit over an empty diagram still exists, it is just the initial object $\emptyset$ of $SC$.

To check the gluing condition, first note that every open $U\subseteq X$ must nontrivially intersect $\Ran^n(M)\times \R_{\geqslant 0}$, the top stratum (in the point-counting stratification). So for $W = U\cap V$, if we have $\alpha\in \mathcal F(U)$ and $\beta \in \mathcal F(V)$ such that $\alpha|_W = \beta|_W$ is a $k$-simplex, then $\alpha$ and $\beta$ must have been $k$-simplices as well. This is because a simplicial takes a simplex to a simplex, and we cannot collide points while remaining in the top stratum. Hence the pullback of $S\owns \alpha$ and $T\owns \beta$ via some induced maps (by descending paths) from $U$ to $W$ and $V$ to $W$, respectively, will restrict to the identity on the chosen $k$-simplex. Hence the gluing condition holds, and $\mathcal F$ is a sheaf. $\square$

Functoriality of $[\ \cdot\ ]$ allows us to extend the proof to build a sheaf valued in simplicial sets.

Proposition 2: Let $\mathcal G$ be the function $\Op(X)^{op}\to \sSet$ on objects given by \[ \mathcal G(U) = \colim\left([\sigma(U)]\rightrightarrows S\ :\ \text{every }[\sigma(U)]\to S \text{ is induced by a descending }\gamma:I\to U\right). \] This is a functor and satisfies the sheaf gluing conditions.

Remark: The sheaf $\mathcal G$ is non-trivial on more sets. For example, any path contained within one stratum of $X$ induces the identity map on simplicial sets (though not on simplicial complexes). Hence $\mathcal G$ is non-trivial on every open set contained within a single stratum.

References: nLab (article "Simplicial complexes"), n-category Cafe (post "Simplicial Sets vs. Simplicial Complexes," 2017-08-19)

Perspectives on the Ran space

This post combines the finite subset approach with the mapping space approach of the Ran space, in the context of stratifications. The goal is to understand the colimit construction of the Ran space, as that leads to more powerful results.

Topology

Let $X,Y$ be topological spaces.

Definition: The mapping space of $X$ with respect to $Y$ is the topological space $X^Y = \{f:Y\to X$ continuous$\}$. The topology on $X^Y$ is the compact-open topology which has as basis finite intersections of sets
\[\{f\in X^Y\ :\ f(K)\subseteq U\}, \hspace{3cm} (1)\]
for all $K\subseteq Y$ compact and all $U\subseteq X$ open.

Now fix a positive integer $n$.

Definition: The Ran space of $X$ is the space $\Ran^{\leqslant n}(X) = \{P\subseteq X\ :\ 0<|P|\leqslant n\}$. The topology on $\Ran^{\leqslant n}(X)$ is the coarsest which contains
\[\left\{P\in \Ran^{\leqslant n}(X)\ :\ P\subseteq \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i \right\} \hspace{3cm} (2)\]
as open sets, for all nonempty finite collection of parwise disjoint open sets $\{U_i\}_{i=1}^k$ in $X$.

From now on, we let $I$ be a set of size $n$ and $M$ be a compact, smooth, connected $m$-manifold. There is a natural map
\[\begin{array}{r c l}
\varphi\ :\ M^I & \to & \Ran^{\leqslant n}(M), \\
(f:I\to M) & \mapsto & f(I).
\end{array}\]
This map is surjective, and for $n>1$, is not injective.

Proposition 1: The map $\varphi$ is continuous and an open map.

Proof: For continuity, take an open set $U\subseteq \Ran^{\leqslant n}(M)$ as in (2) and consider $\varphi^{-1}(U)$. We use the fact that $\{*\}\subset I$ is a compact (in fact open and closed) subset of $I$ and that all the $U_i$ are open, as is their union. Observe that
\begin{align*}\varphi^{-1}(U) & = \left\{f\in M^I \ :\ f(I)\subset \bigcup_{i=1}^k U_i,\ f(I)\cap U_i\neq \emptyset\ \forall\ i \right\} \\
& = \left\{f\in M^I\ :\ f(I)\subset \bigcup_{i=1}^k U_i\right\} \cap \bigcap_{i=1}^k \left\{f\in M^I\ :\ f(*\in I) \in U_i\right\},\end{align*}
which is a finite intersection of sets of the type (1), and so $\varphi^{-1}(U)$ is open in $M^I$.

For openness, take an open set $V$ as in (1), so $V = \bigcap_{i=1}^k \{f\in M^I\ :\ f(K)\subseteq U_i\}$ for different subsets $K\subseteq I$. By Lemma 1, we may assume that the $U_i$ are pairwise disjoint. For each $U_i$, let $\{U_{i,j}\}_{j=1}^\infty$ be a sequence of increasing open sets in $U_i$ such that $U_{i,j}\subseteq U_{i,j+1}$ and $U_{i,j}\xrightarrow{\ j\to\infty\ } U_i$. Then
\[\varphi (V) = \underbrace{\left\{P\in M\ :\ P\subset \bigcup_{i=1}^k U_i,\ P\cap U_i\neq \emptyset\ \forall\ i\right\}}_{f\in M^I\text{\ with image completely in the\ }U_i} \cup \underbrace{\bigcap_{i=1}^k \bigcup_{j=1}^\infty \left\{ P\in M\ :\ P\subset U_{i,j}\cup \left(\overline{U_{i,j}}\right)^c,\ P\cap U_{i,j}\neq \emptyset,\ P \subset \left(\overline{U_{i,j}}\right)^c \neq \emptyset\right\}}_{f\in M^I\text{\ with image partially in the\ }U_i}.\]
Note that $U_{i,j}$ and $\left(\overline{U_{i,j}}\right)^c$, the complement of the closure of $U_{i,j}$ are both open and disjoint in $M$. Since infinite unions and finite intersections of elements in the topology are also open, we have that $\varphi(V)$ is open in $\Ran^{\leqslant n}(M)$. $\square$

The above proposition says that we may talk equivalently about the compact-open topology on $M^I$ and the Ran space topology on $\Ran^{\leqslant n}(M)$. Viewing the Ran space as a function space allows for more general terminology to be applied.

Lemma 1: Let $U_i\subseteq M$ be open, for $i=1,\dots,k$. Then $\bigcap_{i=1}^k\{f\in M^I\ :\ f(K)\subseteq U_i\}$ may be written as a union of intersections $\bigcap_{j=1}^\ell \{f\in M^I\ :\ f(K)\subseteq V_j\}$ with the $V_j$ open, pairwise disjoint, and $\ell\leqslant k$.

Proof: It suffices to prove this in the case $k=2$. Let $U,V\subseteq M$ open and suppose than $U\cap V\neq \emptyset$. Note that $U\setminus V$ and $V\setminus U$ are separated (that is, $(U\setminus V) \cap \overline{V\setminus U} = \emptyset$ and $(V\setminus U)\cap \overline{U\setminus V} = \emptyset$), and since $\R^N$ is a completely normal space (equivalently, satisfies the $T5$ axiom), there exist disjoint open sets $A,B$ with $U\setminus V\subseteq A$ and $U\setminus V\subseteq B$. So for $A' = A\cap (U\cup V)$ and $B' = B\cap (U\cup V)$, we have
\begin{align*} \{f\in M^I\ &:\ f(K)\subseteq U\} \cap \{f\in M^I\ :\ f(K)\subseteq V\} \\
& = \left(\{f\in M^I\ :\ f(K)\subseteq U\setminus V\} \cap \{f\in M^I\ :\ f(K)\subseteq V\setminus U\}\right) \cup \{f\in M^I\ :\ f(K)\subseteq U\cap V \} \\
& = \left(\{f\in M^I\ :\ f(K)\subseteq A'\} \cap \{f\in M^I\ :\ f(K)\subseteq B'\}\right) \cup \{f\in M^I\ :\ f(K)\subseteq U\cap V \}, \end{align*}
for $A',B',U\cap V$ open, and $A'\cap B' = \emptyset$.  $\square$

Note that in the last calculation of the proof, the intersection of sets in the second line is smaller than the intersection of sets in the last line (as $U\setminus V \subsetneq A$ and $V\setminus U\subsetneq B$). However, all the extra ones in the third line appear in the set $\{f\in M^I\ :\ f(K)\subseteq U\cap V\}$.

Stratifications

Now we compare stratifications on $M^I$ and $\Ran^{\leqslant n}(M)$. As before, $I$ is a set of size $n$.

Corollary: An image-constant $A$-stratification on $M^I$ is equivalent to an $A$-stratification on $\Ran^{\leqslant n}(M)$.

This follows from Proposition 1. By image-constant we mean if $\alpha,\beta\in M^I$ have the same image (that is, $\alpha(I)=\beta(I)$), then $\alpha,\beta$ are sent to the same element of $A$.

Proof: If we start with a continuous map $f:M^I\to A$, setting $g(P) = f(I\to M)$ whenever $(I\to M) \in \varphi^{-1}(P)$ is continuous, as $\varphi(f^{-1}(U))$ is open, by continuity of $f$ and openness of $\varphi$. The assignment $g(P) = f(I\to M)$ whenever $(I\to M) \in \varphi^{-1}(P)$ is well defined, as the stratification is image-constant, so any continuous map from $M^I$ must send every element of $\varphi^{-1}(P)$ to the same place.

Conversely, if we start with a continuous map $g:\Ran^{\leqslant n}(M)\to A$, setting $f(I\to M) = g(\varphi(I\to M))$ is continuous, as $\varphi^{-1}(g^{-1}(U))$ is open, by continuity of $g$ and continuity of $\varphi$. This map is image-constant, as $\varphi(\alpha:I\to M) = \alpha(I)$. $\square$

Next we consider a particular stratification of $M^I$, adapted from Example 3.5.17 of Ayala-Francis-Tanaka, simplified with $P=\{*\}$. That is, the example begins with a stratified space $M\to P$ and proceeds to construct another stratification $M^I\to P'$, but we only consider the trivial stratification $M\to \{*\}$.

Definition: Given $M$ and $I$, let the poset $\mathcal P(I)$ of coincidences on $I$ be the set of equivalence relations on $I$, ordered by reverse set inclusion. Let $f_I:M^I\to \mathcal P(I)$ be the natural stratification that takes a map $\alpha: I\to M$ to the equivalence relation on $I$ describing which elements of $I$ coincide in the image of $\alpha$.

Example: An element of $\mathcal P(I)$ is a subset of $I\times I$ always containing $(a,a)$ for every $a\in I$ (reflexivity), and satisfying the symmetry and transitivity conditions. For example, if $|I|=3$ or 4, then $\mathcal P(I)$ is ordered as in the diagrams below, with order increasing from left to right. We simplify things by writing $[x_1,\dots,x_k]$ for the collection $(x_i,x_j)$ of all $i\neq j$ (the equivalence class).

To check that the map $f_I:M^I\to \mathcal P(I)$ is continuous, we first note that an element $U_{[x_1],\dots,[x_k]}$ in the basis of the upwards-directed topology on $\mathcal P(I)$ contains images of $\alpha\in M^I$ whose images have at most the elements of each equivalence class $[x_i]$ coinciding. Hence
\[ f_I^{-1}(U_{[x_1],\dots,[x_k]}) = \bigcup_{U_1,\dots,U_k\subseteq M \atop \text{open, disjoint}}\ \bigcap_{i=1}^k \left\{\alpha\in M^I\ :\ \alpha(K = \{x\in [x_i]\}) \subseteq U_i\right\},\]
which is an open set in the compact-open topolgy on $M^I$.

The Ran space as a colimit

Beilinson-Drinfeld (Section 3.4) and Ayala-Francis-Tanaka (Section 3.7) describe the Ran space as a colimit, the former of a functor into topological spaces, the latter of a functor into stratified spaces. See Mac Lane for a full treatment of colimits. Both BD and AFT use the category $\Fin^{surj,\leqslant n}$ of finite sets and surjections, that is,
\begin{align*}\Obj(\Fin^{surj,\leqslant n}) & = \{I\in \Obj(\Set)\ :\ 0<|I|\leqslant n\}, \\
\Hom_{\Fin^{surj,\leqslant n}}(I,J) & = \begin{cases}
\emptyset, & \text{\ if\ } |I|<|J|, \\
\left\{\text{surjections\ }I\to J\right\}, & \text{\ if\ } |I|\geqslant |J|.
\end{cases}\end{align*}
AFT uses more involved terminology, with "conically smooth" stratified spaces instead of just poset-stratified. They use a category $\Strat$, which for our purposes we may define as
\begin{align*}
\Obj(\Strat) & = \{\text{poset-stratified topological spaces }X\xrightarrow{ f } A\}, \\
\Hom_{\Strat}(X\xrightarrow{ f } A,Y\xrightarrow{ g } B) & = \{(\mu\in \Hom_{\Top}(X,Y), \nu \in \Hom_{\Set}(A,B)\ :\ g\circ \mu = \nu\circ f\}.
\end{align*}

Remark:  There is a natural functor $\mathcal F_M:(\Fin^{surj,\leqslant n})^{op} \to \Top$, given by $I\mapsto M^I$. A surjection $s:I\to J$ induces a map $M^J\to M^I$, with $(f:J\to M)\mapsto (f\circ s :I\to M)$. BD use this to declare that $\Ran^{\leqslant n}(M) = \colim(\mathcal F_M)$.

Remark: There is also a natural functor $\mathcal G_M:(\Fin^{surj,\leqslant n})^{op} \to \Strat$, given by $I\mapsto (M^I\to \mathcal P(I))$. AFT use this to declare that $(\Ran^{\leqslant n}(M)\to \{1,\dots,n\}) = \colim(\mathcal G_M)$.

The construction of AFT is even more general, as they consider the Ran space of an already stratified space. Here we use their result for $M\to \{*\}$ trivially stratified.

References: Ayala, Francis, and Tanaka (Local structures on stratified spaces, Sections 3.5 and 3.7), Beilinson and Drinfeld (Chiral algebras, Section 3.4), Mac Lane (Categories for the working mathematician, Chapter III.3)

Examples of limits and colimits

 Lecture topic

Let $C$ be a category and $X,Y,Z\in \Obj(C)$. Choose $I$ to be a category with $\mathcal F:I\to C$ a functor as described below. Then we may consider the limit and colimit of $\mathcal F$, noting that they may not always exist, as there may be no suitable natural transformation $i$ or $\pi$.

The limit and colimit of the category $I$ with two points and two arrows going between the points in opposite directions, namely

are not interesting to consider. That is because as a category, it must satisfy compositions, so $f\circ g=\id$, which is a restrictive condition on $f$ and $g$. We may define a new map $h:X\to X$ with $h=f\circ g$, but then more maps, such as $h\circ f$ and so on need to be defined, which complicate the situation.

References: Borceux (Handbook of Categorical Algebra I, Chapter 2)

Limits and colimits

 Lecture topic

Definition: Given categories $A,B$ and functors $\mathcal F,\mathcal G:A\to B$, a natural transformation $\eta:\mathcal F\to \mathcal G$ is a collection of elements $\eta_X\in \Hom_B(\mathcal F(X),\mathcal G(X))$ for all $X\in \Obj(A)$ such that the diagram

commutes, whenever $f\in \Hom_A(X,Y)$.

Definition: For $X\in \Obj(A)$, define the constant category $\underline X$ to be the category with $\Obj(\underline X)=\{X\}$ and $\Hom_{\underline X}(X,X)=\{\id_X\}$. For any other category $B$, this may also be viewed as a natural transformation $\underline X:B\to A$ with $\underline X(Y)=X$ and $\underline X(f)=\id_X$ for any object $Y$ and any morphism $f$ of $B$.

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The colimit $\text{colim}(\mathcal F$) of $\mathcal F$ is an object $\text{colim}(\mathcal F)\in \Obj(B)$ and a natural transformation $\iota:\mathcal F\to \underline{\text{colim}(\mathcal F)}$ that is initial among all such natural transformations. We write $\iota_X:\mathcal F(X)\to \text{colim}(\mathcal F)$ and have $\iota(f)=\id_{\text{colim}(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\eta:\mathcal F\to \underline{Z}$ is a natural transformation, there is a unique map $\zeta:\text{colim}(\mathcal F)\to Z$ such that the following diagram commutes:

Definition: Let $A$ be a small category and $\mathcal F:A\to B$ a functor. The limit $\lim(\mathcal F$) of $\mathcal F$ is an object $\lim(\mathcal F)\in \Obj(B)$ and a natural transformation $\pi:\underline{\lim(\mathcal F)}\to \mathcal F$ that is final among all such natural transformations. We write $\pi_X:\lim(\mathcal F) \to \mathcal F(X)$ and have $\pi(f)=\id_{\lim(\mathcal F)}$ for any morphism $f$ of $A$.

In other words, whenever $Z\in \Obj(B)$ and $\epsilon:\underline{Z}\to \mathcal F$ is a natural transformation, there is a unique map $\theta:Z\to \lim(\mathcal F)$ such that the following diagram commutes:

Examples of colimits are initial objects, coproducts, cokernels, pushouts, direct limits. Examples of limits are final objects, products, kernels, pullbacks, inverse limits.

 Remark: $\Hom$ commutes with limits and tensor commutes with colimits. That is:
\[
\Hom(A,\lim(B_i)) = \lim\left(\Hom(A,B_i)\right)
\hspace{1cm}
(\text{colim}(A_i))\otimes B = \text{colim}(A_i\otimes B)
\]
References: May (A Concise course in Algebraic Topology, Chapter 2.6), Aluffi (Algebra: Chapter 0, Chapter VIII.1)